Does economics make progress? The hoary old question was a perennial for many years. You don't hear it so much any more. One reason is a steady accumulation of practical developments. A good example is the modern monetary system.

In a few short years in the late 20th century, plastic cards with magnetic stripes replaced folding money in consumers' wallets for most purposes and the transition from hard money to fiat money was complete.

Bookkeeping entries in bank computers, carefully guaranteed and linked by an intricate web of telecommunications apparatus reliably supported a global economy of unimaginable complexity. And the days when coins contained precious metals seemed very far way.

Yet it was only in 1964 that the US stopped minting coins containing 90 percent silver. Nearly 15 billion circulating dimes, quarters, half- and silver dollars (face value $2.6 billion) then quickly disappeared from circulation in the late 1960s, melted down for ingots by (felonious) entrepreneurs or squirreled away by collectors as the price of silver rose during the inflation associated with the Vietnam War.

By 1970 they were gone.

Just as quickly -- and virtually unnoticed by almost everybody who wasn't minting or melting money at the time -- they had been replaced by today's "clad" coins, composed mostly of copper and zinc, whose intrinsic value as commodity metal is far less than their value in trade. As coins overnight became mere tokens, their hoarding finally ceased altogether in the US. Periodic buying and selling of household silver and gold has continued to be an important fact of life in India and Saudi Arabia to the present day.

It is easy to forget that experience with coins, not banks, drove monetary policy for many centuries. Combining different metals of varying relative values in one unified system was government's central monetary problem for hundreds of years.

The evolution of the general principles governing an orderly coinage that today we take for granted is the subject of an ingenious new book by Thomas Sargent and Francois Velde, The Big Problem of Small Change. The book describes the gradual discovery, through a lengthy process of trial and error, of the "standard formula" by which almost all governments today regulate their currency and coins.

And what is that standard formula? Relatively simple, say Sargent and Velde. Monetary authorities set the various prices, then maintain total convertibility among denominations at fixed rates -- four quarters to one dollar, two dimes and a nickel to one quarter. The public then chooses the quantities of money to be supplied.

Government also regulates the overall quantity of "base money" -- the total amount of government-issued currency and coin in circulation -- with a view to keeping the general level of prices fairly stable over time. These days oversight takes the form of sophisticated manipulation of bank reserves and short-term lending rates, as well as various open-market operations, buying and selling its reserves.

Today's recipe for monetary order is simple only in contrast to the way it used to be. Until about 1850, the system generally worked the other way around. Coins of various denominations consisted of so many lumps of precious metal of different types. Markets, not governments, set the exchange rates among them.

The result was more or less continuous crises -- shortages of coins one decade, rapid changes in relative prices among precious metals the next, large flows of coins across national borders the decade after.

It was these problems that the standard approach to was slowly devised to address. And by the beginning of the 20th century, it did,

An odd topic for a top theorist? Sargent is a professor of economics at Stanford University and a senior fellow of the Hoover Institution there. He was a major contributor to the remarkable but recondite debates about the significance of government credibility for the conduct of monetary policy that took place among university economists during the 1970s and '80s.

There is nothing odd about his latest venture, though, when you when you recall the title Sargent's previous book, The Conquest of American Inflation, which appeared in 1999. Then he was concerned with explaining and evaluating the effort in which he had participated to build a rigorous foundation beneath the age-old insight into the conduct of monetary policy whose gist often has been summarized in the punchline of a familiar joke.

Fool me once, shame on you. Fool me twice, shame on me.

Since the 1980s, monetary theory mainly has had to do with the design of frameworks that might shield central bankers from the highly human temptation to play the "fooling game." The idea is to induce them to rein in more effectively on inflation.

In The Big Problem of Small Change, Sargent undertakes to put in historical perspective the lengthy process by which the steadily growing economies of the West learned to successfully conduct a different policy, namely the vast improvement on the full-bodied money (meaning the gold standard, for example) that today we call fiat money.

Fiat, from the Latin phrase meaning "Let it be done," translates loosely when applied to money as "Trust me, it's worth what it say it is." In truth, the story of the rise of paper money -- now plastic and digital money -- makes a fascinating reading.

The book began in a friendly argument between Sargent and his friend Velde, a senior economist at the Federal Reserve Bank of Chicago who is also a lecturer at the University of Chicago.

Carlo Cippola, the great economic historian of the Middle Ages, had described to an audience the centuries-long process of trial and error that had led to the modern system. Sargent argued that authorities failed at the task of supplying small change because they didn't understand the process. Velde countered that the problem surely was that policy makers in earlier times lacked a reliable means of making coins that were hard to counterfeit.

Thus began a beguiling project. Sargent and Velde braid together strands from the history of ideas, of technology, and of monetary policy -- but with an important further twist. To make their argument as strong as possible, they employ throughout the book a mathematical model, adapted from a Robert Lucas 1982 paper, "Interest Rates and Currency Prices in a Two-Country World."

The idea is to underscore the choices confronting policy makers at every turn. The algebra's demands far exceed the effort that most economists -- and presumably all of us non-economists -- are willing or able to make.

The episodes discussed could hardly be more colorful and diverse -- the sudden expansion of Venetian trade through the issue of lightweight torneselli in Greece in the 14th century, the Kipper- und Wipperzeit ("the clipping and culling times" in 17th century Germany, the debate over England's Great Recoinage of 1696.

The great learning episodes about the power of fiat money seem to have come during periods of siege, especially in Castille and Catalonia, when authorities were forced to make copper do the work of gold. They discovered that money with no intrinsic value could work just as well as precious metal.

The application of formal reasoning to these episodes is itself a reassuring demonstration that history applies to math every bit as much as math applies to history. The use of the model as an engine for thinking is itself is a large part of the effort to demonstrate how learning takes place in economics.

I don't know exactly who can be expected to read this book. Certainly serious students of economics. Reviewers with more time and space to convey the arguments than I have here. But perhaps some vacationers will enjoy it, too, at least those who already have an interest in the history of banking.

The Big Problem of Small Change is a beautiful book. Its approach to the history of money is that of a pair of very stylish detectives working on building a case. The memory of hard money itself exercises a certain fascination. And the way Sargent and Velde deploy the now-standard apparatus of technical economics seems as rhetorically powerful as a siege gun.

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